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Theoretical Analysis and Performance of OFDM Signals in Nonlinear AWGN Channels Paolo Banelli, Member, IEEE, and Saverio Cacopardi, Member, IEEE

Abstract—Orthogonal frequency-division multiplexing (OFDM) baseband signals may be modeled by complex Gaussian processes with Rayleigh envelope distribution and uniform phase distribution, if the number of carriers is sufficiently large. The output correlation function of instantaneous nonlinear amplifiers and the signal-to-distortion ratio can be derived and expressed in an easy way. As a consequence, the output spectrum and the bit-error rate (BER) performance of OFDM systems in nonlinear additive white Gaussian noise channels are predictable both for uncompensated amplitude modulation/amplitude modulation (AM/AM) and amplitude modulation/pulse modulation (AM/PM) distortions and for ideal predistortion. The aim of this work is to obtain the analytical expressions for the output correlation function of a nonlinear device and for the BER performance. The results in closed-form solutions are derived for AM/AM and AM/PM curves approximated by Bessel series expansion and for the ideal predistortion case. Index Terms—Communication system nonlinearities, complex gaussian processes, nonlinear distortions, OFDM, predistortion.

I. INTRODUCTION

A

DSL (asymmetric digital subscriber line) [1], DAB (digital audio broadcasting) [2], and DVB-T (digital video broadcasting-terrestrial) [3] are emerging telecommunication systems that make use of the orthogonal frequency-division multiplexing (OFDM) technique. The OFDM signal, carried by a large number of carriers, is very sensitive to nonlinear distortions because of its greatly variable envelope, which depends on the instantaneous phase value of each carrier. Due to the central limit theorem, the complex baseband OFDM signal can be modeled (for a high number of independently modulated carriers) like a complex Gaussian process with Rayleigh envelope distribution. This allows the analytical treatment of nonlinear OFDM systems making use of the more general results for nonlinear distortions of Gaussian signals. Analog-to-digital (A/D) converters, mixers, and power amplifiers are usually the major sources of nonlinear distortions due to the limited range that they allow for signal dynamics. It is possible to distinguish between two different classes of nonlinear distortions: the first, hereafter named Cartesian, acts separately on the baseband components of the complex signal, while the

Paper approved by B. L. Hughes, the Editor for Theory and Systems of the IEEE Communications Society. Manuscript received August 24, 1998; revised August 1, 1999. The authors are with the Department of Electronic and Information Engineering, University of Perugia, 06126 Perugia, Italy (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0090-6778(00)02272-8.

second acts on the envelope of the complex signal. A/D distortions, called Cartesian clipping, belong to the first class, while AM/AM (amplitude distortion which depends on the amplitude of the input) and AM/PM (phase distortion which depends on the amplitude of the input) introduced by power amplifiers belong to the second class. In the following sections, we will concentrate our analysis on nonlinear distortions introduced by power amplifiers on Gaussian signals, in order to obtain an analytical expression of the output correlation function for envelope nonlinear devices. By using the Fourier transformation, the output correlation function can provide information on the output power spectral density (PSD), and at the same time, it allows the analytical calculation of the output signal-to-nonlinear noise ratio. In this way, it is possible to evaluate the system degradation both for adjacent channel spectral regrowth and bit-error rate (BER) performance in additive white Gaussian noise (AWGN) channels. Original results will be shown for AM/AM and AM/PM nonlinearities expressed by Bessel series expansion and for the envelope clipping, which represents the ideal precorrection situation. It is worth bearing in mind that the results hold for any input signal that can be modeled as a complex zero-mean Gaussian process (e.g. downlink signals for multiuser CDMA systems). II. THEORETICAL BACKGROUND The statistical properties of signals that pass through nonlinear devices have been widely investigated in the past [4]–[11]. In [8], the signal-to-noise ratio (SNR) at the output of real nonlinear devices is derived by making use of the Bussgang theorem. The Bussgang theorem for real functions [12] gives the separateness of a nonlinear output as the sum of a useful attenuated input replica and an uncorrelated nonlinear distortion, expressed by (1) being the input signal and and , respecwith tively, the useful and distorted part of the output signal The same conclusions can be extended to complex nonlinear distortions represented by the combination of AM/AM and AM/PM nonlinearity [9]. is a complex nonlinear distortion funcIf of the input signal tion, which only depends on the envelope then the output signal can be expressed by (2). and

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this paper, we employ the direct method, based on the classical definition of the output correlation function, by which (see Appendix I) it is possible to prove for any nonlinear distortion function that expression (2) can be reduced to Fig. 1.

Nonlinear device in AWGN channels.

, respectively, represent the AM/AM and the AM/PM nonis the nonlinear input–output relationship, linearity, while as pointed out in Fig. 1.

(6) with

expressed by

(7) (2) The complex version of the Bussghang theorem allows expressing the output correlation function by (3). (3) The complex α attenuation coefficient of the useful component is given by

(4) denotes the complex input–output cross-correwhere represents the nonlinear distorting funclation function, the input signal power, the Rayleigh probability tion, the stadensity function (pdf) of the input envelope, and tistical expectation operator. It is thus possible to compute the output signal-to-nonlinear noise ratio as the power ratio between the two uncorrelated output components by the following expression:

where and

is the domain of integration is the Laguerre function expressed by (8)

Expressions (6) and (7) were also derived in [5] in relation to the decomposition of nonlinear outputs as sum of uncorrelated terms. The application of (6) and (7) to nonlinear distortion of an OFDM signal proposed in this paper is new, as well as the closed form solution of the integral in (7) when the nonlinear is expressed by Bessel series expansion or when distortion it is reduced to the soft-limiting of the signal envelope. It is noteworthy that expression (6) represents the expansion equal to the attenuation of expression (3), with of the useful signal power and the series representing the corfor the nonlinear distortion noise. relation function This expression has a simple physical interpretation. In fact, the Fourier transform of the th term in the series represents the part obtained by a th convoluof the output PSD tion of the input PSD with itself, as in the case of a deterministic signal spectrum at the output of an odd nonlinearity, and expressed by

(9)

(5) However, (5) is not the correct expression of the signal-tononlinear noise ratio that degrades the system performance, beis distributed over a cause the nonlinear distortion noise Moreover, this approach wider bandwidth than the signal does not permit derivation of the output correlation function and the PSD for the output signal. Different approaches may be used to compute the output correlation function for nonlinear distortions of Gaussian input signals, including the direct method, the transform method, or the derivative method [10], [11]. In

is the Fourier transform operator and where quency variable.

the fre-

III. OFDM PERFORMANCE IN NONLINEAR AWGN CHANNEL The uncoded system performance in AWGN channels is generally expressed by (10) where SNR is the signal-to-noise ratio at the receiver input. The ratio by the following expression: SNR is related to the (11)

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Fig. 2. Equivalent nonlinear AWGN channels for Gaussian signal.

where is the energy per bit in a symbol period, is the are the bits per symbol PSD of a white Gaussian noise and transmitted. The complex baseband samples of an OFDM signal is transmitted during an OFDM symbol duration represents the expressed by (12) [13] where the unmodulated complex information symbols, the number of angular frequency of the th carrier, and carriers.

same way as the Gaussian noise term obtained by DFT of the thermal Gaussian noise of the receiver [14]. As a consequence, the system performance of an OFDM system can be derived, making use of the same relation used for AWGN channels with the attention of redefining the effective SNR at the receiver input taking into account the nonlinear distortion noise contribution. The nonlinear distortion noise is uncorrelated with the thermal noise; they can be added in power to estimate the BER performance by (10).

(15) It is easy to show (see Appendix IV), by means of (14) that and the useful power term the noise power term are expressed, for each carrier, by (15) and (16).

(12)

(16)

transmitted over each carrier are mapped The symbols on a complex domain (M-QAM, M-DPSK, M-FSK) which is imposed by the application (DAB, DVB-T, WLAN, ADSL, etc). The complex signal received through the nonlinear AWGN channel represented in Fig. 2 is expressed by

and represent the PSD of where represents the number of the the corresponding signals, carriers and switched-on carriers within the total, are the switched-off carriers which represent the guard band of the OFDM system. The two powers expressed by (15) and (16), respectively, represent the sampling on the th carrier and and of position of the PSD of the noise terms in (13). the useful term From the above consideration, it should be clear that the BER performance of the th subchannel associated to the th carrier of the OFDM signal depends both on the nonlinear distortion noise and on the thermal noise. The SNR at the receiver for the th carrier, after the FFT processing, is generally deand fined as the power ratio between the received signal This SNR will be named "apparent" in the thermal noise the following because it is not the only one that influences the performance. It is, however, the measurable one at the receiver and consequently the one in respect of which the BER is generally sketched and the one that is imposed in the simulations. (SNR) is the effective SNR that establishes the performance at the receiver in presence of nonlinear distortions, and it is defined as the power ratio between the useful signal and the total noise after the FFT processing. (SNR) is the power ratio at the receiver in absence of the thermal noise, and it is defined as the power ratio between the useful signal and the nonlinear noise The above quantities are expressed in the (17)–(19) for the generic th carrier of the OFDM system.

(13) The symbols received on the th carrier are expressed, after the discrete Fourier transform (DFT) processing, by

(14) term in (14) is the useful one (i.e., the undistorted The data) when the phase of the α complex coefficient is opportunely compensated at the receiver. This common phase rotation introduced on the constellation of each carrier by the AM/PM curve (through the α coefficient) is not influent in DAB systems because they make use of a differential phase modulation like π/4 and terms, obtained by DFT shifted-DQPSK. The and of the AWGN of the distortion noise contribution are two uncorrelated noise contributions contribution contribution is obthat distort the received symbols. The tained by linear combination of samples of a stationary process. It can be modeled as a complex Gaussian process, because of the central limit theorem, if the number of the distortion errors duration of each that occurs in the time-domain during the distorOFDM symbol is high enough. In this sense, the tion affects the constellation mapped over each carrier, in the

(17)

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of the function for the power amplifier that introduces nonlinear distortions, as expressed in (22) or (B.1)

(22) (18)

and , respectively, the AM/AM and the AM/PM with coefficients can be derived by fitting distortion curves. The procedures on the measured data [16]. The γ term in (22) is the input-backoff and represents the ratio between the input saturation power and the input mean power as expressed by (23), where is the input amplitude of the maximum amplifier output is the signal input power. power and (23)

(19)

coefficients are obtained by substituting (22) in (7) The and are expressed by (24).

and are used to indicate respectively the power and the PSD of the ‘x’ signal. The calculation of the above quantity nonlinearity by (6) and (7). depends on the The uncoded error probability of (10) can be derived for a DAB system bearing in mind that the symbol-error rate (SER) for a single-carrier π/4 Sshifted-DQPSK mapping in the AWGN channel is expressed by [21]

(24)

(20) is the first-order Marcum function and is where the th-order modified Bessel function. In multicarrier systems like DAB, the mean BER for all the carriers is obtained by

nonlinearity, through The coefficients depend on the coefficients of the Bessel expansion (22), and on the γ the coefficients (see Appendix input-backoff, through the II). The Bessel series expansion of the complex nonlinear characteristic of a power amplifier allows to correctly represent almost any kind of amplifier, by using an appropriate number of coefficients to make the fitting error as low as possible. The Saleh model [19] is another typical representation of the complex nonlinear function of a power amplifier. This model has gained a lot of popularity because it represents the complex nonlinear function by an easier analytical notation as expressed by (25) (25)

(21) which represents the average of the BER values calculated on , the carriers (with ) which are used to transmit data. The values of (SER) for each carrier, in presence of nonlinear distortions, is calculated by (20), making use of (19) to express the effective SNR that establishes the system performance. IV. NONLINEAR AMPLIFIERS AND IDEAL PREDISTORTION In this section, we are interested in obtaining an easily manageable expression for the coefficients of (7) when represents the nonlinear distortion introduced by an instantaneous power amplifier with and without ideal predistortion. The Bessel series expansion is a typical representation [15]–[18]

or equivalently (26) where only four coefficients are needed to fit the complex nonlinear function of the amplifier by minimizing the root-meansquare error. Compared to the Bessel series expansion, the Saleh model can be applied to a narrower class of amplifiers, characterized by regular shapes of the AM/AM and AM/PM curves like the ones of typical TWT’s (traveling waves tubes). Nevertheless, the approach presented in this paper does not depend on the amplifier model. As a consequence, if the Saleh model is used to represent the amplifier, the coefficients can be derived by substituting (25) or (26) in (7). Unfortunately, an analytical solution of such integral is not available at this moment, and consequently it must be numerically evaluated.

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Otherwise, is reduced to a real function expressed by (27) if the RF amplifier is ideally predistorted

TABLE I FIRST VALUES OF THE RECURSIVE COEFFICIENTS

(27) In fact, the AM/PM can be completely cancelled by an ideal predistorter, while the AM/AM can only be inverted as far as the saturation power. In this case, the ideally predistorted amplifier exhibits a residual AM/AM that acts as a soft-limiter on the input envelope [16]. In this situation, it is possible to coefficients by means of an exact expression, express the through the following relations [(28)] (see Appendix III) where is the complementary error function, that is

(28)

(a)

The coefficients in (28) are recursively obtained by (29) (see Appendix III)

(29) The first values of the Table I.

coefficient in (29) are outlined in

V. SIMULATION RESULTS The performance comparison between analytical results and computer simulation is demonstrated in this section, as a confirmation of the analytical approach. The DAB system has been chosen as a reference, but the conclusions are extensible to any size is large enough to satisfy OFDM system as long as the the Gaussian signal assumption. The DAB, as with most OFDM systems, uses a cyclic extension of the inverse fast Fourier transform output in order to compensate for multipath at the receiver. This aspect is not considered in our analysis estimating the OFDM PSD because it does not impact on the system performance in nonlinear AWGN channels. In the simulation approach, the PSD was estimated, by means of periodogram, as the average of the PSD (calculated by DFT) of each interpolated (by four) OFDM block. As a consequence, each OFDM

(b) Fig. 3. Measured (a) AM/AM and (b) AM/PM representation by Bessel series expansion.

block shows a quasi-rectangular PSD without significant power contribution in the guard-band region. The PSD obtained in this way is not comparable with the one measured by a spectrum analyzer in a real system, because it does not take into account the phase discontinuities between adjacent OFDM blocks, which cause the spread of the PSD in the adjacent guard-bands. Moreover, the PSD measured by a spectrum analyzer strongly depends on the used video bandwidth, resolution bandwidth, and video averaging. However, it is possible to compare the real measurement with analytical results obtained by the proposed approach, by substituting the PSD measured in linear conditions

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Fig. 4. DAB performance with AM/AM–AM/PM nonlinearities.

at the right-hand side of (9) or, equivalently, substituting its inverse Fourier transform into (6). The DAB signal in the Mode I operating condition [2] is formed by 2048–length blocks characterized by 1 msec of duration (without the cyclic prefix), whose values have been oversampled four times in order to obtain an adequate signal representation in a nonlinear environment. The signal envelope of each interpolated block has been distorted during simulation by the curves (27) and (22) that represent the AM/AM–AM/PM distorting amplifier with and without ideal predistortion, respectively. Both the BER degradation and the spectral regrowth have been evaluated for different γ (input-backoff). Eighty OFDM blocks have been used to estimate the PSD by simulation, in the way previously explained. Fig. 3 shows the AM/AM and coefficients while AM/PM curves fitted by (22) using ten the corresponding obtained results are shown in Fig. 4. Both the PSD regrowth and the corresponding uncoded BER degradation are outlined by using (9) and (21), respectively. Fig. 5 shows the same results when we take into account the ideal predistortion condition expressed by (27). All the analytical expressions used to obtain the system performance depend on the input-backoff (γ) value. However, it is important to esti-

mate the output-backoff (obo) for a meaningful comparison of the predistorted and nonpredistorted conditions. The obo is defined as the ratio between the maximum and the mean output power. It generally depends on both the γ value and on the nonlinear distortion, as expressed by (30) In the ideal predistortion case, the above expression becomes [20] (31) A performance comparison between the predistorted and nonpredistorted conditions is beyond the scope of this paper, as well as considerations about the optimum criterion to fix the in an OFDM system, and it will be the subject of future works. However, an optimum agreement between simulated and analytical results is outlined by Figs. 4 and 5 both for the nonpredistorted and ideally predistorted amplifier when the obo is not dB). For higher obo values, the estoo high (i.e., timated PSD upper-bounds the simulated one. This fact can be explained because OFDM signals are not exactly Gaussian distributed, and their envelope is not perfectly Rayleigh distributed.

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Fig. 5.

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DAB performance with ideal predistortion.

In particular, the maximum envelope will never be infinite, and the accuracy of approximating the pdf with a Rayleigh function is lower for high envelope values. For a certain nonlinear , the increase of the γ value means the reduction distortion of the σ standard deviation of the Gaussian input [see (23)], and as a consequence, the input signal is compressed in the lower curve. region of the Power amplifiers are usually characterized (especially Class curves that are quasi-linear for low input A amplifiers) by values and that localize the distortions in the higher power zone. As a consequence, for high γ values, the nonlinear distortions occur prevalently in correspondence of input envelopes that belong to the right end part of the OFDM pdf where the Rayleigh approximation does not hold anymore. This implies that the coefficient evaluation by (7) is not exact and the resulting PSD does not agree with the simulated one when the γ value is carrier too high. The approximation error is reduced if the number becomes higher, and consequently, the OFDM signal is better approximated by a Gaussian pdf. An ideally predistorted amplifier distorts the signal only for the high input powers after the saturation zone and, as a consequence, is more sensitive to the inadequate analytical representation for high input-backoff. For practical γ values below 9 dB, however, the PSD results tes-

tify to the correctness of the analytical derivation of the coefficients for the output correlation function (6), while the BER results testify to the correctness of the proposed model for nonlinear AWGN channels. Similar considerations can moreover be extended to nonlinear fading channels when the equalization technique at the receiver is considered [16]. Finally, Fig. 6 shows the baseband output PSD for a 2000–carrier OFDM system in linearized environment. Each term’s contribution of (6) to the signal values. spectral re-growth is outlined for two different VI. CONCLUSIONS We have reviewed the analytical background for expressing the output correlation function for complex Gaussian signals distorted by envelope dependent nonlinearities. Original results for AM/AM and AM/PM nonlinearities represented by Bessel series expansion have been derived. Original results are also outlined for the envelope clipping phenomenon which may be compared with the known results for the cartesian clipping [10], [23]–[25]. The envelope clipping is particularly interesting because it represents the ideal predistortion of the AM/AM and AM/PM nonlinearities introduced by high power amplifiers.

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(a)

(b) Fig. 6. Spectral components of an envelope clipped OFDM signal. (a) Ideal predistortion: N carriers = 2000; N carriers = 2000; = 3 dB; obo = 3.63 dB.

These results have been used to propose a model for the analytical performance evaluation of OFDM systems in nonlinear AWGN channels. Computer simulations have validated both the proposed model and the analytical derivations. This approach constitutes a generalization and an improvement of the semianalytical approach already known in the literature [14]. The computer calculation time to predict the OFDM system performance has been dramatically reduced. The proposed approach can also be extended to nonlinear fading channels, in order to outline the OFDM system performance in realistic environments. In this case, the effects of the nonlinear distortion

= 6 dB; obo = 6.08 dB. (b) Ideal predistortion:

noise on the equalizer performance must be taken into account, and this will be the subject of future work. APPENDIX A OUTPUT CORRELATION FUNCTION FOR NONLINEAR DISTORTIONS THAT DEPEND ON COMPLEX GAUSSIAN SIGNAL ENVELOPE Denote (A.1)

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a complex stationary Gaussian process with equal distributed, and components. Under these zero-mean, independent defined hypotheses

With some manipulations, substituting (A.8) and (A.3) into (A.10), it is straightforward to obtain

(A.2) the joint pdf is expressed by (A.11) with The double integral in (A.11) is computable by modified Bessel functions to obtain (A.3) with the autocorrelation coefficient ρ and the input power respectively, expressed by

, (A.12)

(A.4) with

is the modified Bessel function of the first order and where first kind. Making use of the Laguerre polynomial series expansion [22, eq. (5.11.3.7)], expressed by (A.13)

the statistical expectation (A.5) (A.13)

The input correlation function is by definition expressed by

and substituting it into (A.12), the expression of the output correlation function becomes (A.14). (A.6) with The above expression, for a complex process equally distributed and independent real components, becomes (A.7) represents the complex nonlinear distortion function, If then the which only depends on the input envelope of output distorted signal is expressed by

The double integral in (A.14) is separable in respect to to obtain the general expression (A.15).

(A.14) and

(A.8) Based on the above definitions, the output correlation function is given by

(A.15)

(A.9) The integral in (A.9) in polar coordinates becomes

APPENDIX B NONLINEAR AMPLIFIER WITH AM/AM AND AM/PM EXPRESSED BY BESSEL SERIES EXPANSION If is the complex AM/AM and AM/PM distortion function, it can be represented by Bessel series expansion (B.1) are complex coefficients where

(A.10)

(B.1)

BANELLI AND CACOPARDI: OFDM SIGNALS IN NONLINEAR AWGN CHANNELS

The parameter may be chosen equal to (B.2) to be comrepresenting pliant with the notation used in [16], with a normalization factor, the input amthe input power, plitude to which corresponds the saturation power, and γ the input-backoff defined in (23).

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The integrals in (C.2) are all analytically solvable (the first and third by [22, eqs. (2.19.1.1), (2.19.3.5)], respectively), and after some manipulations it is possible to obtain

(B.2) Substituting (B.1) into (10), we obtain the following expression for the coefficients:

(C.3) is the input-backoff in respect to the softwhere limiter saturation point. Expression (C.3) may be enough to calculate easily the coefficient, if attention is paid to truncating the series expansions at a -term high enough to make the error small enough. Anyway, it is possible to analytically calculate the sum of the series contained in (C.3) obtaining an exact closed-form expression for coefficients. the is the th-order polynomial expressed by If

(B.3) The above integral is analytically solvable [22, eq. (2.19.12.6)] and (B.3) becomes

(C.4) and division of

are the quotient and the remainder of the by , respectively, as expressed by

(B.4) (C.5) which coincides with (24). It should be highlighted that the Bessel series representation (B.1) of the amplifier nonlinearity (i.e., on the does not depend on the signal input power γ input-backoff). The PSD on the contrary depends on the γ input-backoff trough, the parameter of (B.2).

The above expression is meaningful only for expression becomes tuting (C.5) into (C.3), the

Substi-

APPENDIX C IDEAL PREDISTORTION (ENVELOPE SOFT-LIMITER) In this case, the nonlinear function expressed by

is a real function

(C.1) coeffiSubstituting (C.1) into (10), the expression of the cients becomes (C.2) with a variable change expressed by (C.6) After some manipulations and making use of known series sums [22], (C.6) becomes (C.7)

(C.2)

(C.7)

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simply obtained by (C.5) and expressed by (C.8) and coefficients that are expressed by (C.9)

(C.15) It is clear from (C.13) that (C.8) (C.16) (C.9) of Now, we need to make coefficient of function of coefficient From (C.5), it is possible to write (C.10)

explicit as a

and by the combination of (C.15), (C.16), (C.12), and (C.7)–(C.9), the recursive algorithm of Section II for the coefficient calculation is obtained. are obtained in an analogous way The values for from expression (C.3). APPENDIX D NOISE POWER ON THE TH CARRIER The receiver input in the time-domain after A/D conversion is expressed by (D.1)

(C.10) by which it is evident that

The received symbol on the th carrier after the DFT processing (see Fig. 1) is expressed by (D.2) by means of (14) and (16).

(C.11) As a consequence, the tained by

coefficients are recursively ob(D.2) The power of the error contribution

is expressed by

(C.12) coefficient of can also be recursively calcuThe lated making use of the definition in (C.4)

(C.13) which can be exploited to obtain (C.14). The identity in (C.14) coeffiallows us to obtain the recursive expression for the cient as expressed by (C.15).

(C.14)

(D.3)

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where is the discrete PSD, the continuous PSD of are the active carriers the noise contribution, and size and the number obtained by the difference between the of the switched-off carriers which constitute the guard-band of the OFDM system. REFERENCES [1] Asymmetric digital subscriber line (ADSL) metallic interface, ANSI T1.413, 1995. [2] Radio broadcast systems; Digital audio broadcasting (DAB) to mobile, portable and fixed receivers., Final Draft prETS 300 401, Nov. 1994. [3] Framing structure, channel coding and modulation for digital terrestrial television, DVB Document A012, June 1996. [4] N. M. Blachman, “Bandpass nonlinearities,” IEEE Trans. Inform. Theory, vol. IT-10, pp. 162–164, Apr. 1964. [5] N. M. Blachman, “The output signals and noise from a nonlinearity with amplitude-dependent phase shift,” IEEE Trans. Inform. Theory, vol. IT-25, pp. 77–79, Jan. 1979. [6] O. Shimbo, “Effects of intermodulation, AM-PM conversion and additive noise in multicarrier TWT systems,” Proc. IEEE, vol. 59, pp. 230–238, Feb. 1971. [7] J. C. Fuenzalida, O. Shimbo, and W. L. Cook, “The domain analysis of intermodulation effects caused by non linear amplifiers,” COMSAT Tech. Rev., vol. 3, no. 1, 1973. [8] H. E. Rowe, “Memoryless nonlinearities with Gaussian inputs: Elementary results,” Bell Syst. Tech. J., vol. 61, no. 7, pp. 1520–1523, Sept. 1982. [9] J. Minkoff, “The role of AM-to-PM conversion in memoryless nonlinear systems,” IEEE Trans. Commun., vol. COM-33, pp. 139–143, Feb. 1985. [10] B. Levine, Fondements theoriques de la radiotechnique statistique. New York: Editions de Moscou, 1973. [11] W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New YorK: McGraw-Hill, 1958. [12] A. Papoulis, Probability, Random Variables and Stochastic Process, 3rd ed. New York: McGraw-Hill, 1991, p. 307. [13] L. Cimini, “Analysis and simulation of a digital mobile channel using orthogonal frequency division multiplexing,” IEEE Trans. Commun., vol. COMM-33, pp. 665–675, 1985. [14] G. Santella and F. Mazzenga, “A model for performance evaluation in M-QAM-OFDM schemes in presence of nonlinear distortions,” in Proc. IEEE Vehicular Technology Conf. (VTC ), Chicago, IL, July 1995, pp. 830–834. [15] N. M. Blachman, “Detectors, bandpass nonlinearities, and their optimization: Inversion of the Chebyshev transform,” IEEE Trans. Inform. Theory, vol. IT-17, pp. 398–404, July 1971. [16] A. R. Kaye, D. A. George, and M. J. Eric, “Analysis and compensation of bandpass nonlinearities for communications,” IEEE Trans. Commun., vol. COM-20, pp. 965–972, Oct. 1972. [17] “Behavioral modeling of narrowband microwave power amplifiers with application in simulating spectral regrowth,” in IEEE Int. Microwave Symp. Dig., vol. 3, 1996, pp. 1385–1388.

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[18] G. Chrisikos and C. J. Clark et al., “A nonlinear ARMA model for simulating power amplifiers,” in IEEE Int. Microwave Symp. Dig., vol. 2, 1998, pp. 733–736. [19] A. Saleh, “Frequency-independent and frequency-dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun., vol. 29, pp. 1715–1720, Nov. 1981. [20] S. Cacopardi, P. Banelli, F. Frescura, and G. Reali, “OM-DS-SS wireless LAN radio subsystem: Performance in clipping environment using measured channel delay profiles,” in IEEE GLOBECOM, London, U.K., Nov. 1996, pp. 1897–1903. [21] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [22] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. New York: Gordon and Breach, 1986. [23] J. Rinne and M. Renfors, “The behavior of orthogonal frequency division multiplexing signals in amplitude limiting channel,” in Proc. IEEE Int. Conf. Communications (ICC), New Orleans, LA, 1994, pp. 381–385. [24] R. O’Neill and L. B. Lopes, “Performance of amplitude limited multitone signals,” in Proc. IEEE Vehicular Technology Conf. (VTC), Stockholm, Sweden, June 1994, pp. 1675–1679. [25] Q. Shi, “OFDM in bandpass nonlinearity,” IEEE Trans. Consumer Electron., vol. 42, pp. 253–258, Aug. 1996.

Paolo Banelli (S’90–M’99) was born in Perugia, Italy, on May 19, 1968. He received the Laurea degree in electronics engineering and the Ph.D.degree in telecommunications from the University of Perugia, Perugia, Italy, in 1993 and 1998, respectively. Currently, he is an Assistant Professor with the Department of Electronic and Information Engineering at the University of Perugia. His research interests include nonlinear distortions, broadcasting, and mobile communications.

Saverio Cacopardi (M’83) was born in Torino, Italy, on October 23, 1941. He received the Laurea degree in electrical engineering from the University of Rome, Rome, Italy, in 1970. From 1971 to 1975, he worked on PCM transmission and CATV at SIP (Italian Telephone Operating Company). In 1975, he joined the Electrical Communication Department at the University of Rome as an Assistant Professor. From 1979 to 1985, he was an Associate Professor of Electrical Communications at the University of Ancona. From 1986 to 1991, he was an Associate Professor of Radio Aids to Navigation at the University "La Sapienza," Rome, Itay. In 1991, he joined the University of Perugia where he is now an Full Professor of Electrical Communications. His current research interests include mobile communications, WLAN, and broadcasting.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 48, NO. 3, MARCH 2000

Theoretical Analysis and Performance of OFDM Signals in Nonlinear AWGN Channels Paolo Banelli, Member, IEEE, and Saverio Cacopardi, Member, IEEE

Abstract—Orthogonal frequency-division multiplexing (OFDM) baseband signals may be modeled by complex Gaussian processes with Rayleigh envelope distribution and uniform phase distribution, if the number of carriers is sufficiently large. The output correlation function of instantaneous nonlinear amplifiers and the signal-to-distortion ratio can be derived and expressed in an easy way. As a consequence, the output spectrum and the bit-error rate (BER) performance of OFDM systems in nonlinear additive white Gaussian noise channels are predictable both for uncompensated amplitude modulation/amplitude modulation (AM/AM) and amplitude modulation/pulse modulation (AM/PM) distortions and for ideal predistortion. The aim of this work is to obtain the analytical expressions for the output correlation function of a nonlinear device and for the BER performance. The results in closed-form solutions are derived for AM/AM and AM/PM curves approximated by Bessel series expansion and for the ideal predistortion case. Index Terms—Communication system nonlinearities, complex gaussian processes, nonlinear distortions, OFDM, predistortion.

I. INTRODUCTION

A

DSL (asymmetric digital subscriber line) [1], DAB (digital audio broadcasting) [2], and DVB-T (digital video broadcasting-terrestrial) [3] are emerging telecommunication systems that make use of the orthogonal frequency-division multiplexing (OFDM) technique. The OFDM signal, carried by a large number of carriers, is very sensitive to nonlinear distortions because of its greatly variable envelope, which depends on the instantaneous phase value of each carrier. Due to the central limit theorem, the complex baseband OFDM signal can be modeled (for a high number of independently modulated carriers) like a complex Gaussian process with Rayleigh envelope distribution. This allows the analytical treatment of nonlinear OFDM systems making use of the more general results for nonlinear distortions of Gaussian signals. Analog-to-digital (A/D) converters, mixers, and power amplifiers are usually the major sources of nonlinear distortions due to the limited range that they allow for signal dynamics. It is possible to distinguish between two different classes of nonlinear distortions: the first, hereafter named Cartesian, acts separately on the baseband components of the complex signal, while the

Paper approved by B. L. Hughes, the Editor for Theory and Systems of the IEEE Communications Society. Manuscript received August 24, 1998; revised August 1, 1999. The authors are with the Department of Electronic and Information Engineering, University of Perugia, 06126 Perugia, Italy (e-mail: [email protected]; [email protected]). Publisher Item Identifier S 0090-6778(00)02272-8.

second acts on the envelope of the complex signal. A/D distortions, called Cartesian clipping, belong to the first class, while AM/AM (amplitude distortion which depends on the amplitude of the input) and AM/PM (phase distortion which depends on the amplitude of the input) introduced by power amplifiers belong to the second class. In the following sections, we will concentrate our analysis on nonlinear distortions introduced by power amplifiers on Gaussian signals, in order to obtain an analytical expression of the output correlation function for envelope nonlinear devices. By using the Fourier transformation, the output correlation function can provide information on the output power spectral density (PSD), and at the same time, it allows the analytical calculation of the output signal-to-nonlinear noise ratio. In this way, it is possible to evaluate the system degradation both for adjacent channel spectral regrowth and bit-error rate (BER) performance in additive white Gaussian noise (AWGN) channels. Original results will be shown for AM/AM and AM/PM nonlinearities expressed by Bessel series expansion and for the envelope clipping, which represents the ideal precorrection situation. It is worth bearing in mind that the results hold for any input signal that can be modeled as a complex zero-mean Gaussian process (e.g. downlink signals for multiuser CDMA systems). II. THEORETICAL BACKGROUND The statistical properties of signals that pass through nonlinear devices have been widely investigated in the past [4]–[11]. In [8], the signal-to-noise ratio (SNR) at the output of real nonlinear devices is derived by making use of the Bussgang theorem. The Bussgang theorem for real functions [12] gives the separateness of a nonlinear output as the sum of a useful attenuated input replica and an uncorrelated nonlinear distortion, expressed by (1) being the input signal and and , respecwith tively, the useful and distorted part of the output signal The same conclusions can be extended to complex nonlinear distortions represented by the combination of AM/AM and AM/PM nonlinearity [9]. is a complex nonlinear distortion funcIf of the input signal tion, which only depends on the envelope then the output signal can be expressed by (2). and

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this paper, we employ the direct method, based on the classical definition of the output correlation function, by which (see Appendix I) it is possible to prove for any nonlinear distortion function that expression (2) can be reduced to Fig. 1.

Nonlinear device in AWGN channels.

, respectively, represent the AM/AM and the AM/PM nonis the nonlinear input–output relationship, linearity, while as pointed out in Fig. 1.

(6) with

expressed by

(7) (2) The complex version of the Bussghang theorem allows expressing the output correlation function by (3). (3) The complex α attenuation coefficient of the useful component is given by

(4) denotes the complex input–output cross-correwhere represents the nonlinear distorting funclation function, the input signal power, the Rayleigh probability tion, the stadensity function (pdf) of the input envelope, and tistical expectation operator. It is thus possible to compute the output signal-to-nonlinear noise ratio as the power ratio between the two uncorrelated output components by the following expression:

where and

is the domain of integration is the Laguerre function expressed by (8)

Expressions (6) and (7) were also derived in [5] in relation to the decomposition of nonlinear outputs as sum of uncorrelated terms. The application of (6) and (7) to nonlinear distortion of an OFDM signal proposed in this paper is new, as well as the closed form solution of the integral in (7) when the nonlinear is expressed by Bessel series expansion or when distortion it is reduced to the soft-limiting of the signal envelope. It is noteworthy that expression (6) represents the expansion equal to the attenuation of expression (3), with of the useful signal power and the series representing the corfor the nonlinear distortion noise. relation function This expression has a simple physical interpretation. In fact, the Fourier transform of the th term in the series represents the part obtained by a th convoluof the output PSD tion of the input PSD with itself, as in the case of a deterministic signal spectrum at the output of an odd nonlinearity, and expressed by

(9)

(5) However, (5) is not the correct expression of the signal-tononlinear noise ratio that degrades the system performance, beis distributed over a cause the nonlinear distortion noise Moreover, this approach wider bandwidth than the signal does not permit derivation of the output correlation function and the PSD for the output signal. Different approaches may be used to compute the output correlation function for nonlinear distortions of Gaussian input signals, including the direct method, the transform method, or the derivative method [10], [11]. In

is the Fourier transform operator and where quency variable.

the fre-

III. OFDM PERFORMANCE IN NONLINEAR AWGN CHANNEL The uncoded system performance in AWGN channels is generally expressed by (10) where SNR is the signal-to-noise ratio at the receiver input. The ratio by the following expression: SNR is related to the (11)

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Fig. 2. Equivalent nonlinear AWGN channels for Gaussian signal.

where is the energy per bit in a symbol period, is the are the bits per symbol PSD of a white Gaussian noise and transmitted. The complex baseband samples of an OFDM signal is transmitted during an OFDM symbol duration represents the expressed by (12) [13] where the unmodulated complex information symbols, the number of angular frequency of the th carrier, and carriers.

same way as the Gaussian noise term obtained by DFT of the thermal Gaussian noise of the receiver [14]. As a consequence, the system performance of an OFDM system can be derived, making use of the same relation used for AWGN channels with the attention of redefining the effective SNR at the receiver input taking into account the nonlinear distortion noise contribution. The nonlinear distortion noise is uncorrelated with the thermal noise; they can be added in power to estimate the BER performance by (10).

(15) It is easy to show (see Appendix IV), by means of (14) that and the useful power term the noise power term are expressed, for each carrier, by (15) and (16).

(12)

(16)

transmitted over each carrier are mapped The symbols on a complex domain (M-QAM, M-DPSK, M-FSK) which is imposed by the application (DAB, DVB-T, WLAN, ADSL, etc). The complex signal received through the nonlinear AWGN channel represented in Fig. 2 is expressed by

and represent the PSD of where represents the number of the the corresponding signals, carriers and switched-on carriers within the total, are the switched-off carriers which represent the guard band of the OFDM system. The two powers expressed by (15) and (16), respectively, represent the sampling on the th carrier and and of position of the PSD of the noise terms in (13). the useful term From the above consideration, it should be clear that the BER performance of the th subchannel associated to the th carrier of the OFDM signal depends both on the nonlinear distortion noise and on the thermal noise. The SNR at the receiver for the th carrier, after the FFT processing, is generally deand fined as the power ratio between the received signal This SNR will be named "apparent" in the thermal noise the following because it is not the only one that influences the performance. It is, however, the measurable one at the receiver and consequently the one in respect of which the BER is generally sketched and the one that is imposed in the simulations. (SNR) is the effective SNR that establishes the performance at the receiver in presence of nonlinear distortions, and it is defined as the power ratio between the useful signal and the total noise after the FFT processing. (SNR) is the power ratio at the receiver in absence of the thermal noise, and it is defined as the power ratio between the useful signal and the nonlinear noise The above quantities are expressed in the (17)–(19) for the generic th carrier of the OFDM system.

(13) The symbols received on the th carrier are expressed, after the discrete Fourier transform (DFT) processing, by

(14) term in (14) is the useful one (i.e., the undistorted The data) when the phase of the α complex coefficient is opportunely compensated at the receiver. This common phase rotation introduced on the constellation of each carrier by the AM/PM curve (through the α coefficient) is not influent in DAB systems because they make use of a differential phase modulation like π/4 and terms, obtained by DFT shifted-DQPSK. The and of the AWGN of the distortion noise contribution are two uncorrelated noise contributions contribution contribution is obthat distort the received symbols. The tained by linear combination of samples of a stationary process. It can be modeled as a complex Gaussian process, because of the central limit theorem, if the number of the distortion errors duration of each that occurs in the time-domain during the distorOFDM symbol is high enough. In this sense, the tion affects the constellation mapped over each carrier, in the

(17)

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of the function for the power amplifier that introduces nonlinear distortions, as expressed in (22) or (B.1)

(22) (18)

and , respectively, the AM/AM and the AM/PM with coefficients can be derived by fitting distortion curves. The procedures on the measured data [16]. The γ term in (22) is the input-backoff and represents the ratio between the input saturation power and the input mean power as expressed by (23), where is the input amplitude of the maximum amplifier output is the signal input power. power and (23)

(19)

coefficients are obtained by substituting (22) in (7) The and are expressed by (24).

and are used to indicate respectively the power and the PSD of the ‘x’ signal. The calculation of the above quantity nonlinearity by (6) and (7). depends on the The uncoded error probability of (10) can be derived for a DAB system bearing in mind that the symbol-error rate (SER) for a single-carrier π/4 Sshifted-DQPSK mapping in the AWGN channel is expressed by [21]

(24)

(20) is the first-order Marcum function and is where the th-order modified Bessel function. In multicarrier systems like DAB, the mean BER for all the carriers is obtained by

nonlinearity, through The coefficients depend on the coefficients of the Bessel expansion (22), and on the γ the coefficients (see Appendix input-backoff, through the II). The Bessel series expansion of the complex nonlinear characteristic of a power amplifier allows to correctly represent almost any kind of amplifier, by using an appropriate number of coefficients to make the fitting error as low as possible. The Saleh model [19] is another typical representation of the complex nonlinear function of a power amplifier. This model has gained a lot of popularity because it represents the complex nonlinear function by an easier analytical notation as expressed by (25) (25)

(21) which represents the average of the BER values calculated on , the carriers (with ) which are used to transmit data. The values of (SER) for each carrier, in presence of nonlinear distortions, is calculated by (20), making use of (19) to express the effective SNR that establishes the system performance. IV. NONLINEAR AMPLIFIERS AND IDEAL PREDISTORTION In this section, we are interested in obtaining an easily manageable expression for the coefficients of (7) when represents the nonlinear distortion introduced by an instantaneous power amplifier with and without ideal predistortion. The Bessel series expansion is a typical representation [15]–[18]

or equivalently (26) where only four coefficients are needed to fit the complex nonlinear function of the amplifier by minimizing the root-meansquare error. Compared to the Bessel series expansion, the Saleh model can be applied to a narrower class of amplifiers, characterized by regular shapes of the AM/AM and AM/PM curves like the ones of typical TWT’s (traveling waves tubes). Nevertheless, the approach presented in this paper does not depend on the amplifier model. As a consequence, if the Saleh model is used to represent the amplifier, the coefficients can be derived by substituting (25) or (26) in (7). Unfortunately, an analytical solution of such integral is not available at this moment, and consequently it must be numerically evaluated.

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Otherwise, is reduced to a real function expressed by (27) if the RF amplifier is ideally predistorted

TABLE I FIRST VALUES OF THE RECURSIVE COEFFICIENTS

(27) In fact, the AM/PM can be completely cancelled by an ideal predistorter, while the AM/AM can only be inverted as far as the saturation power. In this case, the ideally predistorted amplifier exhibits a residual AM/AM that acts as a soft-limiter on the input envelope [16]. In this situation, it is possible to coefficients by means of an exact expression, express the through the following relations [(28)] (see Appendix III) where is the complementary error function, that is

(28)

(a)

The coefficients in (28) are recursively obtained by (29) (see Appendix III)

(29) The first values of the Table I.

coefficient in (29) are outlined in

V. SIMULATION RESULTS The performance comparison between analytical results and computer simulation is demonstrated in this section, as a confirmation of the analytical approach. The DAB system has been chosen as a reference, but the conclusions are extensible to any size is large enough to satisfy OFDM system as long as the the Gaussian signal assumption. The DAB, as with most OFDM systems, uses a cyclic extension of the inverse fast Fourier transform output in order to compensate for multipath at the receiver. This aspect is not considered in our analysis estimating the OFDM PSD because it does not impact on the system performance in nonlinear AWGN channels. In the simulation approach, the PSD was estimated, by means of periodogram, as the average of the PSD (calculated by DFT) of each interpolated (by four) OFDM block. As a consequence, each OFDM

(b) Fig. 3. Measured (a) AM/AM and (b) AM/PM representation by Bessel series expansion.

block shows a quasi-rectangular PSD without significant power contribution in the guard-band region. The PSD obtained in this way is not comparable with the one measured by a spectrum analyzer in a real system, because it does not take into account the phase discontinuities between adjacent OFDM blocks, which cause the spread of the PSD in the adjacent guard-bands. Moreover, the PSD measured by a spectrum analyzer strongly depends on the used video bandwidth, resolution bandwidth, and video averaging. However, it is possible to compare the real measurement with analytical results obtained by the proposed approach, by substituting the PSD measured in linear conditions

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Fig. 4. DAB performance with AM/AM–AM/PM nonlinearities.

at the right-hand side of (9) or, equivalently, substituting its inverse Fourier transform into (6). The DAB signal in the Mode I operating condition [2] is formed by 2048–length blocks characterized by 1 msec of duration (without the cyclic prefix), whose values have been oversampled four times in order to obtain an adequate signal representation in a nonlinear environment. The signal envelope of each interpolated block has been distorted during simulation by the curves (27) and (22) that represent the AM/AM–AM/PM distorting amplifier with and without ideal predistortion, respectively. Both the BER degradation and the spectral regrowth have been evaluated for different γ (input-backoff). Eighty OFDM blocks have been used to estimate the PSD by simulation, in the way previously explained. Fig. 3 shows the AM/AM and coefficients while AM/PM curves fitted by (22) using ten the corresponding obtained results are shown in Fig. 4. Both the PSD regrowth and the corresponding uncoded BER degradation are outlined by using (9) and (21), respectively. Fig. 5 shows the same results when we take into account the ideal predistortion condition expressed by (27). All the analytical expressions used to obtain the system performance depend on the input-backoff (γ) value. However, it is important to esti-

mate the output-backoff (obo) for a meaningful comparison of the predistorted and nonpredistorted conditions. The obo is defined as the ratio between the maximum and the mean output power. It generally depends on both the γ value and on the nonlinear distortion, as expressed by (30) In the ideal predistortion case, the above expression becomes [20] (31) A performance comparison between the predistorted and nonpredistorted conditions is beyond the scope of this paper, as well as considerations about the optimum criterion to fix the in an OFDM system, and it will be the subject of future works. However, an optimum agreement between simulated and analytical results is outlined by Figs. 4 and 5 both for the nonpredistorted and ideally predistorted amplifier when the obo is not dB). For higher obo values, the estoo high (i.e., timated PSD upper-bounds the simulated one. This fact can be explained because OFDM signals are not exactly Gaussian distributed, and their envelope is not perfectly Rayleigh distributed.

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Fig. 5.

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DAB performance with ideal predistortion.

In particular, the maximum envelope will never be infinite, and the accuracy of approximating the pdf with a Rayleigh function is lower for high envelope values. For a certain nonlinear , the increase of the γ value means the reduction distortion of the σ standard deviation of the Gaussian input [see (23)], and as a consequence, the input signal is compressed in the lower curve. region of the Power amplifiers are usually characterized (especially Class curves that are quasi-linear for low input A amplifiers) by values and that localize the distortions in the higher power zone. As a consequence, for high γ values, the nonlinear distortions occur prevalently in correspondence of input envelopes that belong to the right end part of the OFDM pdf where the Rayleigh approximation does not hold anymore. This implies that the coefficient evaluation by (7) is not exact and the resulting PSD does not agree with the simulated one when the γ value is carrier too high. The approximation error is reduced if the number becomes higher, and consequently, the OFDM signal is better approximated by a Gaussian pdf. An ideally predistorted amplifier distorts the signal only for the high input powers after the saturation zone and, as a consequence, is more sensitive to the inadequate analytical representation for high input-backoff. For practical γ values below 9 dB, however, the PSD results tes-

tify to the correctness of the analytical derivation of the coefficients for the output correlation function (6), while the BER results testify to the correctness of the proposed model for nonlinear AWGN channels. Similar considerations can moreover be extended to nonlinear fading channels when the equalization technique at the receiver is considered [16]. Finally, Fig. 6 shows the baseband output PSD for a 2000–carrier OFDM system in linearized environment. Each term’s contribution of (6) to the signal values. spectral re-growth is outlined for two different VI. CONCLUSIONS We have reviewed the analytical background for expressing the output correlation function for complex Gaussian signals distorted by envelope dependent nonlinearities. Original results for AM/AM and AM/PM nonlinearities represented by Bessel series expansion have been derived. Original results are also outlined for the envelope clipping phenomenon which may be compared with the known results for the cartesian clipping [10], [23]–[25]. The envelope clipping is particularly interesting because it represents the ideal predistortion of the AM/AM and AM/PM nonlinearities introduced by high power amplifiers.

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(a)

(b) Fig. 6. Spectral components of an envelope clipped OFDM signal. (a) Ideal predistortion: N carriers = 2000; N carriers = 2000; = 3 dB; obo = 3.63 dB.

These results have been used to propose a model for the analytical performance evaluation of OFDM systems in nonlinear AWGN channels. Computer simulations have validated both the proposed model and the analytical derivations. This approach constitutes a generalization and an improvement of the semianalytical approach already known in the literature [14]. The computer calculation time to predict the OFDM system performance has been dramatically reduced. The proposed approach can also be extended to nonlinear fading channels, in order to outline the OFDM system performance in realistic environments. In this case, the effects of the nonlinear distortion

= 6 dB; obo = 6.08 dB. (b) Ideal predistortion:

noise on the equalizer performance must be taken into account, and this will be the subject of future work. APPENDIX A OUTPUT CORRELATION FUNCTION FOR NONLINEAR DISTORTIONS THAT DEPEND ON COMPLEX GAUSSIAN SIGNAL ENVELOPE Denote (A.1)

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a complex stationary Gaussian process with equal distributed, and components. Under these zero-mean, independent defined hypotheses

With some manipulations, substituting (A.8) and (A.3) into (A.10), it is straightforward to obtain

(A.2) the joint pdf is expressed by (A.11) with The double integral in (A.11) is computable by modified Bessel functions to obtain (A.3) with the autocorrelation coefficient ρ and the input power respectively, expressed by

, (A.12)

(A.4) with

is the modified Bessel function of the first order and where first kind. Making use of the Laguerre polynomial series expansion [22, eq. (5.11.3.7)], expressed by (A.13)

the statistical expectation (A.5) (A.13)

The input correlation function is by definition expressed by

and substituting it into (A.12), the expression of the output correlation function becomes (A.14). (A.6) with The above expression, for a complex process equally distributed and independent real components, becomes (A.7) represents the complex nonlinear distortion function, If then the which only depends on the input envelope of output distorted signal is expressed by

The double integral in (A.14) is separable in respect to to obtain the general expression (A.15).

(A.14) and

(A.8) Based on the above definitions, the output correlation function is given by

(A.15)

(A.9) The integral in (A.9) in polar coordinates becomes

APPENDIX B NONLINEAR AMPLIFIER WITH AM/AM AND AM/PM EXPRESSED BY BESSEL SERIES EXPANSION If is the complex AM/AM and AM/PM distortion function, it can be represented by Bessel series expansion (B.1) are complex coefficients where

(A.10)

(B.1)

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The parameter may be chosen equal to (B.2) to be comrepresenting pliant with the notation used in [16], with a normalization factor, the input amthe input power, plitude to which corresponds the saturation power, and γ the input-backoff defined in (23).

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The integrals in (C.2) are all analytically solvable (the first and third by [22, eqs. (2.19.1.1), (2.19.3.5)], respectively), and after some manipulations it is possible to obtain

(B.2) Substituting (B.1) into (10), we obtain the following expression for the coefficients:

(C.3) is the input-backoff in respect to the softwhere limiter saturation point. Expression (C.3) may be enough to calculate easily the coefficient, if attention is paid to truncating the series expansions at a -term high enough to make the error small enough. Anyway, it is possible to analytically calculate the sum of the series contained in (C.3) obtaining an exact closed-form expression for coefficients. the is the th-order polynomial expressed by If

(B.3) The above integral is analytically solvable [22, eq. (2.19.12.6)] and (B.3) becomes

(C.4) and division of

are the quotient and the remainder of the by , respectively, as expressed by

(B.4) (C.5) which coincides with (24). It should be highlighted that the Bessel series representation (B.1) of the amplifier nonlinearity (i.e., on the does not depend on the signal input power γ input-backoff). The PSD on the contrary depends on the γ input-backoff trough, the parameter of (B.2).

The above expression is meaningful only for expression becomes tuting (C.5) into (C.3), the

Substi-

APPENDIX C IDEAL PREDISTORTION (ENVELOPE SOFT-LIMITER) In this case, the nonlinear function expressed by

is a real function

(C.1) coeffiSubstituting (C.1) into (10), the expression of the cients becomes (C.2) with a variable change expressed by (C.6) After some manipulations and making use of known series sums [22], (C.6) becomes (C.7)

(C.2)

(C.7)

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simply obtained by (C.5) and expressed by (C.8) and coefficients that are expressed by (C.9)

(C.15) It is clear from (C.13) that (C.8) (C.16) (C.9) of Now, we need to make coefficient of function of coefficient From (C.5), it is possible to write (C.10)

explicit as a

and by the combination of (C.15), (C.16), (C.12), and (C.7)–(C.9), the recursive algorithm of Section II for the coefficient calculation is obtained. are obtained in an analogous way The values for from expression (C.3). APPENDIX D NOISE POWER ON THE TH CARRIER The receiver input in the time-domain after A/D conversion is expressed by (D.1)

(C.10) by which it is evident that

The received symbol on the th carrier after the DFT processing (see Fig. 1) is expressed by (D.2) by means of (14) and (16).

(C.11) As a consequence, the tained by

coefficients are recursively ob(D.2) The power of the error contribution

is expressed by

(C.12) coefficient of can also be recursively calcuThe lated making use of the definition in (C.4)

(C.13) which can be exploited to obtain (C.14). The identity in (C.14) coeffiallows us to obtain the recursive expression for the cient as expressed by (C.15).

(C.14)

(D.3)

BANELLI AND CACOPARDI: OFDM SIGNALS IN NONLINEAR AWGN CHANNELS

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Paolo Banelli (S’90–M’99) was born in Perugia, Italy, on May 19, 1968. He received the Laurea degree in electronics engineering and the Ph.D.degree in telecommunications from the University of Perugia, Perugia, Italy, in 1993 and 1998, respectively. Currently, he is an Assistant Professor with the Department of Electronic and Information Engineering at the University of Perugia. His research interests include nonlinear distortions, broadcasting, and mobile communications.

Saverio Cacopardi (M’83) was born in Torino, Italy, on October 23, 1941. He received the Laurea degree in electrical engineering from the University of Rome, Rome, Italy, in 1970. From 1971 to 1975, he worked on PCM transmission and CATV at SIP (Italian Telephone Operating Company). In 1975, he joined the Electrical Communication Department at the University of Rome as an Assistant Professor. From 1979 to 1985, he was an Associate Professor of Electrical Communications at the University of Ancona. From 1986 to 1991, he was an Associate Professor of Radio Aids to Navigation at the University "La Sapienza," Rome, Itay. In 1991, he joined the University of Perugia where he is now an Full Professor of Electrical Communications. His current research interests include mobile communications, WLAN, and broadcasting.